A thin rod of length L is lying along the x-axis with its ends at x=0 and x=L its linear (mass/length) varies with x as `k(x/L)^n`, where n can be zero of any positive number. If to position `x_(CM)` of the centre of mass of the rod is plotted against 'n', which of the following graphs best apporximates the dependence of `x_(CM)` on n?

Options

A.
B.
C.
D.

Answer

A

Explanation

Position of centre of mass.
`X_(CM) = (int_0^L(k/(L^n) .x^n .dx)x)/(int_0^Lk/(L_n) . X^n . Dx) = (int__0^L x^(n+1)dx)/(int_0^L x^ndx) = (L^(n+2))/(n +2) cdot ((n+1))/(L^(n+1)) implies X_(CM) = (L(n + 1))/((n +2))`
The variation of the centre of mass with x is given by
`(dx)/(dn) = L{((n+2)1 - (n + 1))/((n + 2)^2)} = L/((n+2)^2)`
If the rod has the same density as x = 0, i.e., n = 0, therefore uniform, the centre of mass would have been at L/2. As the density increases with length, the centre of mass shifts towards the right. Therefore it can only be option (a).